This preview shows pages 1–2. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: 18.03 Problem Set 4: Solutions 13. (a) The system is underdamped; the roots are not real. (b) Since the system is underdamped, the solutions are of the form x = Ae bt/ 2 cos( d t ). Then x = Ae bt/ 2 (( b/ 2) cos( d t ) + d sin( d t )). The term following the exponential is sinusoidal, with circular frequency d . Therefore it vanishes every / d seconds, and this is the time between a maximum and a minimum, or half of a cycle. We have discovered that / d = 4 / 2 = 2, or d = 2 radians/sec. Since the roots of the characteristic polynomial are b 2 d i , we have discovered that their imaginary parts are / 2. (c) For any t , A cos( d t ) = A cos( d ( t +4) ), since d = 2 . So the ratio x ( t +4) /x ( t ) is given by the ratio of the other terms: x ( t + 4) /x ( t ) = e b ( t +4) / 2 /e bt/ 2 = e 2 b . This tells us that 1 / 2 = e 2 b , or b = (ln 2) / 2: The real part of the roots is (ln 2) / 4....
View
Full
Document
 Spring '09
 unknown

Click to edit the document details